Abstract

A nonhomogeneous universe with vacuum energy, but without
spacetime expansion, is utilized together with gravitational and Doppler
redshifts as the basis for proposing a new interpretation of the Hubble
relation and the 2.7K Cosmic Blackbody Radiation.

Hans-Dieter Radecke recently noted1
that modern cosmology's dependence
on "interpretations of interpretations of observations" means that "We
should not to fall victim to cosmological hubris, but stay open for any surprise."
We now report what seems a major surprise—namely, the discovery of
a New Redshift Interpretation (NRI) of the Hubble redshift relation and 2.7K
Cosmic Blackbody Radiation (CBR) without assuming either the Friedmann-Lemaitre
wavelength expansion hypothesis or the Cosmological Principle, the
latter being long acknowledged as the ". . . one great uncertainty that hangs
like a dark cloud over the standard model."2 Whereas the standard model and
the NRI both interpret nearby galactic redshifts as Doppler shifts, they differ
significantly in their interpretation of distant redshifts. This difference can
be traced to two fundamentally different views of the universe. The standard
model utilizes a universe governed by expanding-spacetime general relativity
whereas the NRI is based on a universe governed by static-spacetime
general relativity. A brief review of early twentieth-century astronomy and
cosmology assists in focusing more precisely on the nature of this difference.

In 1917 Einstein applied his newly developed static-spacetime general
theory of relativity to cosmology,3
and introduced a cosmological constant to
maintain the universe in what was then thought to be a static condition. But
Edwin Hubble's momentous 1929 discovery4 that galactic redshifts increase
in proportion to their distance challenged the static universe concept. His
discovery confronted cosmologists with two surprises, and they were initially
unprepared to deal with either. First, they were unaware of any static-spacetime
redshift interpretation which could account for increasing galactic
redshifts in a real, finite-density universe. Secondly, if Hubble's results were
interpreted as Doppler shifts they implied omnidirectional galactic recession,
which in turn implied the existence of a universal center near the Galaxy.

In any event, whatever efforts cosmologists might have put forth to obtain
a static-spacetime interpretation of Hubble's discovery were effectively
cut short when their attention was soon directed to the potential cosmological
implications of the hitherto virtually unnoticed results of Alexander
Friedmann5 and Georges Lemaitre,6
both of whom had found expanding-spacetime
solutions of the Einstein field equations in the early and mid-1920s.
Their results were attractive for two reasons. First, uniform spacetime expansion
showed promise for eliminating the implication of the Galaxy occupying
a preferred position in the universe. Hubble spoke for most cosmologists
of his time when he forthrightly admitted an extreme distaste for such a
possibility, saying it should be accepted only as a last resort.7

Second, Lemaitre hypothesized that, apart from the well-known redshift
due relative motion of source and observer, expanding-spacetime should cause
photons everywhere to experience continuous, in-flight wavelength expansion
proportional to the expansion itself.6
Thus was born the concept of spacetime
expansion redshifts, given by zexp =
ℜo/ℜe − 1; where
ℜo and ℜe
represent the magnitudes of the postulated Friedmann-Lemaitre spacetime expansion
factors at observation and emission.6

Despite its critical role in standard model theory, the foregoing expression
for zexp is unique in that the physical existence of
ℜ has never been
verified by experiment; the reason is that no method has yet been proposed
to measure ℜ; either past or present. Even so, expansion redshifts have become
the cornerstone of the standard model for two reasons—namely, (1)
because the experimentally determined Hubble redshift relation, z = Hr/c,
can be developed as a theoretical consequence of spacetime expansion theory
if the hypothesized expansion redshifts, zexp =
ℜo/ℜe
− 1, are assumed to
be identical with zobs = λo/λe
− 1; the observed redshifts of distant galaxies,
and (2) because of their key role in providing what has previously been
thought to be a unique interpretation of the 2.7K CBR. That interpretation
assumes the much earlier existence of a primeval fireball radiation wherein
matter/radiation decoupling occurred at about 3000K when the expansion
redshift was about 1000 compared to the present. It follows that a 1000-fold
redshifting of such a radiation by spacetime expansion would result in the
presently observed 2.7K CBR. The Hubble relation and 2.7K CBR scenarios
are widely understood as confirming the existence of expansion redshifts.
Such a conclusion may be premature, however, seeing that the crucially important
expansion factor, ℜ; has yet to be experimentally verified.

This brings us to the standard model's second fundamental assumption
known as the Cosmological Principle—namely, that in the large scale the universe
is homogeneous and isotropic, or put in simpler terms, it is everywhere
alike. This Principle was earlier noted to be the ". . . one great uncertainty
that hangs like a dark cloud over the standard model."2
Uncertainty exists
because, even though the Hubble relation is powerful evidence for large-scale
isotropy about the Galaxy, we simply cannot confirm universal homogeneity
because we lack knowing whether the Hubble relation would result if redshift
measurements were made from points of observation on other galaxies.

Nevertheless the standard model requires homogeneity because in it galaxies
are assumed to be comoving bodies in expanding spacetime. That is, since
spacetime expansion is assumed to be uniform, comoving galactic separation
must likewise be uniform, which implies that all observers, regardless of location,
should see the same general picture of the universe. This is what the
standard model requires, and it is observationally unprovable.

In summary, then, our mini-review of twentieth century astronomy and
cosmology have revealed two reasons why we cannot be absolutely certain
of Friedmann-Lemaitre expansion redshifts and the standard model's cornerstone
postulate of a no-center universe governed by expanding-spacetime
general relativity. First, the universal homogeneity required by the standard
model is acknowledged to be observationally unprovable. Second, despite
the fact that in theory all photons in the universe should be synchronously
experiencing in-flight wavelength expansion in direct proportion to the instantaneous
value of ℜ; until now little attention has been given to finding
a method to test this prediction. More on this later. For the present we say
only that the foregoing uncertainties are sufficient to suggest the possibility
that the universe may not be governed by expanding-spacetime general
relativity required by the standard model. As far as is known this paper is
the first attempt to seriously explore the cosmological consequences of such
a possibility and, as will now be seen, the results do appear quite surprising.

The foregoing account provides the basis for understanding why the NRI
attempts to account for the Hubble relation and the 2.7K CBR by using
Doppler and gravitational redshifts embedded in a universe governed by
static-spacetime general relativity. Without expanding spacetime there can
be no Cosmological Principle, and without this Principle the Hubble relation
implies the existence of a center in the NRI. In it the Hubble redshifts are
now interpreted solely in terms of relativistic Doppler and Einstein gravitational
redshifts, all cast within the framework of a finite, nonhomogeneous,
vacuum-gravity universe with cosmic center (C) near the Galaxy.

The NRI framework assumes the widely dispersed galaxies of the visible
universe are enclosed by a thin, outer shell of hot hydrogen at a distance
R from the Galaxy. Thus, the volume of space enclosed by this luminous
shell—assumed, for ease of calculation, to have a nearly uniform temperature
of 5400K—would completely fill with blackbody cavity radiation. But the
radial variation of gravitational potential within this volume means the cavity
radiation temperature measured at any interior point would depend on the
magnitude of the Einstein gravitational redshift between that point and the
outer shell. By including relativistic vacuum energy density, ρv, and pressure,
pv, into the gravitational structure of the cosmos we now show how 5400K
radiation emitted at R could be gravitationally redshifted by a factor of 2000
so as to appear as 2.7K blackbody cavity radiation here at the Galaxy.

In particular, if pv is negative, then, as
Novikov8 shows, ρv will be positive,
and the summed vacuum pressure/energy contributions to vacuum gravity
will be −2ρv. So, excluding the spherical hydrogen shell at R, the net density
throughout the cosmos from C to R would be ρ − 2ρv,
where ρ is the average
mass/energy density. Beyond R both densities are assumed to either cancel
or exponentially diminish to infinitesimal values, which effectively achieves
for the NRI framework what Birkhoff's theorem did for standard cosmology.
This framework is sufficient to compute the gravitational potentials needed
to calculate both Hubble and 2.7K CBR redshifts in the NRI framework.

If Φ(0) and Φ(R) represent the universal potentials at C and R, then,
Φ(R) = −(G/R)[M1 + MS], and
Φ(0) = −G[∫0R4π(ρ
− 2ρv)r · dr + MS/R],
where MS is the mass of the thin, outer shell, and the net mass from C to
R is M1 = 4πR3(ρ
− 2ρv)/3. To find MS we employ the boundary condition
Φ(0) = 0, which first yields MS =
−2πR3(ρ − 2ρv)
and then, by substitution,
the expression Φ(R) = 2πGR2[ρ
− 2ρv]/3. Explanation of Φ(0) = 0 is given
in ref. (9). Gravitationally redshifting the uniform 5400K radiation at R so
as to appear as the 2.7K CBR at C means that TR = 5400K and TC = 2.7K,
in which case the gravitational redshift at C would be,

z + 1 = √1 + 2Φ(0)/c² /
√1 + 2Φ(R)/c² =
TR / TC = 5400/2.7.

(1)

If, as soon to be explained, we let ρv =
8.790 × 10−30 g/cm3 and ρ = 2 × 10−31
g/cm3, then Eq. (1) yields R≅
1.362 × 1028 cm, or about 14.24 × 109 ly.
These are the conditions which allow the 2.7K CBR to be interpreted as
blackbody cavity radiation. Now consider the Hubble relation.

Interpreting the Hubble relation in the NRI framework assumes that distant
galaxies may have both radial and transverse velocity components relative
to C. Thus the appropriate redshift equation is given by the standard
expression,10

z = [1 − v→ ·
k^/c]√1 + 2Φ(0)/c² /
√1 + 2Φ(r)/c² − v²/c² − 1

(2)

which describes how light emitted from a distant galaxy moving at velocity
v→—with the unit vector k^
pointing from source to observer—will be redshifted
by a combination of gravitational and special relativistic Doppler effects. In
Eq. (2) r is the distance from C to an arbitrary galaxy where the gravitational
potential is,

To obtain the Hubble relation from Eq. (4) we note that if 2ρv > ρ, then
the (ρ − 2ρv) density factor
will cause any galaxy located at a distance r from
C to experience an outward radial acceleration, r¨ =
−GM/r2, due to the
enclosed negative mass M = 4πr3(ρ −
2ρv)/3. This leads to the equation
r¨ = br, where b = 4πG(2ρv
− ρ)/3. Its solution is r =
rgexp√bt, where
rg is a galaxy-specific, initial condition parameter. Taking its proper time
derivative leads to the expression vr = r⋅ =
√br, which is of course the Hubble
velocity-distance relation, vr = Hr, with H =
√b.

If ρ = 2 × 10−31g-cm−3
and H = 68 km-s−1-Mpc−1 =
2.203 × 10−18s−1
are substituted into H2 = b = 4πG(2ρv
− ρ)/3, then the vacuum density is ρv≅ 8.79 × 10−30g-cm−3.
This value for H was earlier utilized to calculate
R ≅ 14.24 billion ly. And substitution of this value of the vacuum density
in ρvc2 =
Gmv6c4/ℏ4,
as given by Novikov8, yields the average mass, mv, of
the virtual particles emitted by the vacuum. If me is the electron mass, then
mv≃ 82me
for ρ = 2 × 10−31g-cm−3.

If we consider the case for dark matter, where ρ =
2 × 10−30g-cm−3, then
ρv≅ 9.79 ×
10−30g-cm−3
and mv≃ 83me.
Thus the use of dark matter
density in the NRI produces only negligible changes in mv and ρv, which
has the important consequence of justifying the use of ρv as an adjustable
parameter in the expression
H2 = 4πG(2ρv − ρ)/3.
Indeed, because the right
side of this expression occurs explicitly in the NRI redshift Equation (4), it's
possible to restate it in terms of H2 without a density term. (See further
discussion in the next paragraph.) This means the results already obtained,
as well as those that now follow, apply equally to a universe with or without
the assumption of dark matter.

For small r, Eq. (5) reduces to the Hubble redshift
relation, z = Hr/c,
which means the NRI framework successfully interprets both it and the 2.7K
CBR. Even the latter's microtemperature variations11
can be interpreted as
temperature variations in the 5400K outer shell. Going further, the equation,
r¨ = br, with b as given above, essentially duplicates12
the inflationary expansion relation, ℜ¨
= {8πGρf / 3}ℜ.

The NRI framework also succeeds in interpreting the observed variation
of CBR temperature with redshift. When Songaila et
al.13 investigated this
topic, their measurement yielded a CBR temperature of 7.4 ± 0.8 K for
z = 1.776. The NRI framework's prediction is obtained by substituting r for
R in Eq. (1), from which it follows that the CBR temperature should vary
spatially with r and z as Tz = TC /
√1 + 2Φ(r)/c² = (z + 1)TC.
For TC = 2.726K, then T1.776 = 7.57K,
which agrees with experiment and duplicates
the standard cosmology's prediction, but without employing the expansion
hypothesis, inflationary or otherwise.

Can the NRI framework also be applied to the interpretation of quasar
redshifts? Perhaps so if we focus on the recent confirmation of the paucity
of quasars with z > 4, and their near absence for z >
5.14 If, for example,
we assume ug2≃ 0.5 and vθ = 0.7c
might apply to some of the most distant
quasars, then Eq. (5) reveals that z increases from 4.3 at 8.8 × 109 ly to
534 at 9.1263 × 109 ly. Could such a rapid increase in z be interpreted as
one reason for the sparsity of very high redshift quasars? If so, quasars with
z > 4.3 might have proper motions of ~ 15 μas/y.

Next, consider Hubble Deep Field. It is of interest to note that its analysis
shows that angular diameters of the most distant galaxies do not go through
a minimum and then increase as predicted by the standard cosmology.15
Instead angular diameters continue to diminish. Standard cosmology's prediction
is of course traceable to its underlying assumption of spacetime expansion.
Since the NRI framework does not have that assumption, it can
quite naturally interpret Hubble Deep Field results as evidence of the applicability
of the general astronomical rule-of-thumb for galaxies—the further
the distance, the smaller the angular diameter.

The NRI framework's apparent ability to interpret a variety of astronomical
and astrophysical observations with Einstein's gravitational and Doppler
redshifts, without the addition of the Friedmann-Lemaitre spacetime wavelength
expansion hypothesis and the Cosmological Principle, was such a surprise
that it seemed quite natural to extend this study and reanalyze the
physics underlying the expansion hypothesis itself.

Our ongoing investigations on this topic have led us to uncover an intriguing,
new test of the expansion hypothesis.16 The end result is yet another
surprise almost on par with finding that the NRI framework is an alternate
interpretation of the Hubble relation and the 2.7K CBR.

Gravity in the NRI is the sum of local
and universal potentials. Thus two clocks at distances s1
and s2 from Earth's center should run at rates
that differ by √1 + 2[Φ(0) + φ(s1)]/c² / √1 +
2[Φ(0) + φ(s2)]/c², where φ(s)
is the Earth's local potential. The synchronization of GPS atomic
clocks with the √1 + 2φ(s1)/c² / √1 + 2φ(s2)/c² ratio is consistent with Φ(0)
= 0.